We determined the number of fissions that occur in the development of a hadron shower in a 238U calorimeter, using a method based on the analysis of induced radioactivity. Measurements were done at 300 GeV (π-, and at 591 MeV (protons). The number of fissions turns out to be much smaller (∼ 10 fissions per GeV) than usually assumed, and is very sensitive to the calorimeter configuration. For example, in massive 238U the number of neutron-induced fissions is 25% larger than in a fine-sampling uranium/scintillator device. The results for calorimeters with a high-Z readout are similar to the massive U case. A significant fraction (10-15%) of the fissions are caused by fast charged particles rather than by slow neutrons. We also determined the total neutron production in the proton beam. It turns out that less than 25% of the neutrons produced in the shower development cause fission. The insertion of low-Z readout layers decisively determines how fast and by which mechanisms the neutrons lose their kinetic energy. The neutron flux in lead is about 40% of that in uranium, but the neutrons are on an average faster. We measured that 4.1 ± 0.4 fisions per GeV are on an average induced in the shower development of an incoming 300 GeV π. An attempt is made to explain these data. The consequences for calorimetric measurements of the hadron energy are discussed.